Subgroups — C₂×C₃×ΓL(1,𝔽₈)⋊D₇₁ #31822078

Order 17892.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×ΓL(1,𝔽₈)⋊D₇₁	C₂	1	1	yes	maximal, characteristic, direct product factor
3	C₃×ΓL(1,𝔽₈)⋊D₇₁	C₂	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×D₄₉₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×ΓL(1,𝔽₈)⋊D₇₁	C₃	1	3	yes	maximal, direct product factor
6	C₃×ΓL(1,𝔽₈)×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
7	ΓL(1,𝔽₈)⋊D₇₁	C₂×C₃	1	3	yes	direct product factor
8	ΓL(1,𝔽₈)⋊D₇₁	C₂×C₃	1	3	yes	direct product factor
9	C₃×D₄₉₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×D₄₉₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×ΓL(1,𝔽₈)×C₇₁	C₂×C₃	1	3	yes	complemented
12	C₂×C₃×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃²×D₇₁	—	7	1	no	maximal, Hall
14	C₂×D₄₉₇	C₃²	1	1	yes	characteristic, complemented, Hall
15	C₃×C₇×C₇₁	C₂²×C₃	1	1	yes	characteristic, complemented
16	ΓL(1,𝔽₈)×C₇₁	C₂²×C₃	1	3	yes	complemented
17	C₂×C₃²×C₇₁	—	7	1	no	—
18	C₃²×D₇₁	—	7	1	no	—
19	C₃²×D₇₁	—	7	1	no	—
20	C₂×C₇×C₇₁	C₂×C₃²	1	1	yes	characteristic, complemented
21	D₄₉₇	C₂×C₃²	1	1	yes	characteristic, complemented
22	D₄₉₇	C₂×C₃²	1	1	yes	characteristic, complemented
23	C₂×C₃×D₇₁	—	7	3	no	—
24	C₂×C₃×D₇₁	—	7	1	no	—
25	C₃²×C₇₁	—	7	1	no	Hall
26	C₇×C₇₁	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₇₁	—	7	3	no	—
28	C₂×C₃×C₇₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₃×D₇₁	—	7	3	no	—
30	C₃×D₇₁	—	7	3	no	—
31	C₃×D₇₁	—	21	1	no	—
32	C₃×D₇₁	—	21	1	no	—
33	C₂×D₇₁	—	7	1	no	Hall
34	C₂×C₃×AGL(1,𝔽₇)	—	71	1	no	maximal, Hall
35	C₃×C₇₁	—	7	3	no	—
36	C₃×C₇₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
37	C₂×C₇₁	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
38	D₇₁	—	21	1	no	—
39	D₇₁	—	21	1	no	—
40	C₂×C₃×ΓL(1,𝔽₈)	D₇₁	1	1	yes	characteristic, complemented
41	C₃×AGL(1,𝔽₇)	—	71	1	no	—
42	C₃×AGL(1,𝔽₇)	—	71	1	no	—
43	C₂×C₃×D₇	—	71	1	no	—
44	C₂×AGL(1,𝔽₇)	—	71	3	no	—
45	C₇₁	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
46	C₃×ΓL(1,𝔽₈)	C₂×D₇₁	1	1	yes	characteristic, complemented, Hall
47	C₂×C₃×C₇	C₃×D₇₁	1	1	yes	characteristic, complemented
48	C₃×D₇	—	71	1	no	—
49	C₃×D₇	—	71	1	no	—
50	C₂×ΓL(1,𝔽₈)	C₃×D₇₁	1	3	yes	complemented
51	AGL(1,𝔽₇)	—	71	3	no	—
52	AGL(1,𝔽₇)	—	71	3	no	—
53	C₂²×C₃²	—	497	1	no	Hall
54	C₂×D₇	—	71	1	no	Hall
55	C₃×C₇	C₂×C₃×D₇₁	1	1	yes	characteristic, complemented
56	ΓL(1,𝔽₈)	C₂×C₃×D₇₁	1	3	yes	complemented
57	C₂×C₃²	—	7	1	no	—
58	C₂×C₃²	—	497	1	no	—
59	C₂×C₃²	—	497	1	no	—
60	C₂×C₇	C₃²×D₇₁	1	1	yes	characteristic, complemented
61	D₇	—	71	1	no	—
62	D₇	—	71	1	no	—
63	C₂²×C₃	—	1491	1	no	—
64	C₂²×C₃	—	497	3	no	—
65	C₃²	—	7	1	no	Sylow
66	C₇	C₂×C₃²×D₇₁	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₃	—	7	3	no	—
68	C₂×C₃	—	1491	1	no	—
69	C₂×C₃	—	497	3	no	—
70	C₂×C₃	—	497	3	no	—
71	C₂×C₃	—	497	1	no	—
72	C₂×C₃	ΓL(1,𝔽₈)⋊D₇₁	1	1	yes	characteristic, direct product factor
73	C₂²	—	1491	1	no	Sylow
74	C₃	—	7	3	no	—
75	C₃	C₂×ΓL(1,𝔽₈)⋊D₇₁	1	1	yes	characteristic, direct product factor
76	C₂	—	1491	1	no	—
77	C₂	—	497	1	no	—
78	C₂	C₃×ΓL(1,𝔽₈)⋊D₇₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₈)⋊D₇₁
commutator, cosocle, hypocenter	C₇×C₇₁	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₇₁	C₂×C₃
LCS	G ▷ C₇×C₇₁
UCS	C₁ ◁ C₂×C₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂	semidirect
2. C₃×ΓL(1,𝔽₈)⋊D₇₁	C₂	direct (N × H)
3. C₃×ΓL(1,𝔽₈)⋊D₇₁	C₂	direct (N × H)
4. C₂×C₃×D₄₉₇	C₃	semidirect
5. C₂×ΓL(1,𝔽₈)⋊D₇₁	C₃	direct (N × H)
6. C₃×ΓL(1,𝔽₈)×C₇₁	C₂²	semidirect
7. ΓL(1,𝔽₈)⋊D₇₁	C₂×C₃	direct (N × H)
8. ΓL(1,𝔽₈)⋊D₇₁	C₂×C₃	direct (N × H)
9. C₃×D₄₉₇	C₂×C₃	semidirect
10. C₃×D₄₉₇	C₂×C₃	semidirect
11. C₂×ΓL(1,𝔽₈)×C₇₁	C₂×C₃	semidirect
12. C₂×C₃×C₇×C₇₁	C₂×C₃	semidirect
14. C₂×D₄₉₇	C₃²	semidirect
15. C₃×C₇×C₇₁	C₂²×C₃	semidirect
16. ΓL(1,𝔽₈)×C₇₁	C₂²×C₃	semidirect
20. C₂×C₇×C₇₁	C₂×C₃²	semidirect
21. D₄₉₇	C₂×C₃²	semidirect
22. D₄₉₇	C₂×C₃²	semidirect
26. C₇×C₇₁	C₂²×C₃²	semidirect
28. C₂×C₃×C₇₁	AGL(1,𝔽₇)	semidirect
36. C₃×C₇₁	C₂×AGL(1,𝔽₇)	semidirect
37. C₂×C₇₁	C₃×AGL(1,𝔽₇)	semidirect
40. C₂×C₃×ΓL(1,𝔽₈)	D₇₁	semidirect
45. C₇₁	C₂×C₃×AGL(1,𝔽₇)	semidirect
46. C₃×ΓL(1,𝔽₈)	C₂×D₇₁	semidirect
47. C₂×C₃×C₇	C₃×D₇₁	semidirect
50. C₂×ΓL(1,𝔽₈)	C₃×D₇₁	semidirect
55. C₃×C₇	C₂×C₃×D₇₁	semidirect
56. ΓL(1,𝔽₈)	C₂×C₃×D₇₁	semidirect
60. C₂×C₇	C₃²×D₇₁	semidirect
66. C₇	C₂×C₃²×D₇₁	semidirect
72. C₂×C₃	ΓL(1,𝔽₈)⋊D₇₁	direct (N × H)
75. C₃	C₂×ΓL(1,𝔽₈)⋊D₇₁	direct (N × H)
78. C₂	C₃×ΓL(1,𝔽₈)⋊D₇₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×ΓL(1,𝔽₈)×C₇₁	2
C₃×ΓL(1,𝔽₈)⋊D₇₁	2
C₂×C₃×D₄₉₇	3
C₂×ΓL(1,𝔽₈)⋊D₇₁	3
C₂×C₃²×D₇₁	7
C₂×C₃×AGL(1,𝔽₇)	71

Subgroups — C2×C3×ΓL(1,𝔽8)⋊D71 #31822078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×ΓL(1,𝔽₈)⋊D₇₁ #31822078